Question

Evaluate the following :

$\left|\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right|$

Answer 1

Let $\Delta =\left|\begin{array}{ccc}x+\lambda & x& x\\ x& x+\lambda & x\\ x& x& x+\lambda \end{array}\right|$.

Applying $C_1\to C_1+C_2+C_3,$ we get

$\left|\begin{array}{ccc}3x+\lambda & x& x\\ 3x+\lambda & x+\lambda & x\\ 3x+\lambda & x& x+\lambda \end{array}\right|=(3x+\lambda )\left|\begin{array}{ccc}1& x& x\\ 1& x+\lambda & x\\ 1& x& x+\lambda \end{array}\right|$ [Taking $(3x+\lambda )$ common from $C_1$]

$\Delta =(3x+\lambda )\left|\begin{array}{ccc}1& x& x\\ 0& \lambda & 0\\ 0& 0& \lambda \end{array}\right|$ [Applying $R_2\to R_2-R_1,R_3\to R_3-R_1]$

$\Rightarrow \Delta =(3x+\lambda )\left|\begin{array}{ccc}\lambda & 0& \\ 0& \lambda & \end{array}\right|$ [Expanding along $C_1$]

$\Rightarrow \Delta ={\lambda }^2(3x+\lambda )$